The dynamics of the simplest torus quantum vortex knots in a superfluid at zero temperature has been simulated with a regularized Biot–Savart law (the torus radii R 0 and r 0 for the initial vortex configuration are much larger than the width of the vortex core ). The evolution times of knots until their significant deformation have been calculated with a small step in the parameter B 0 = r 0 / R 0 for different values of the parameter Λ = log( R 0 /ξ). It has been found that regions of quasi-stability appear at Λ ≳ 3 in the range B 0 ≲ 0.2, which correspond to long knot lifetimes and very large traveling distances up to several hundred R 0 . This result is new and quite surprising because previously it was believed that the maximum lifetime of torus knots until reconnection does not exceed several typical periods. The opening of quasi-stable “windows” at increasing Λ is due to narrowing of main parametric resonances of the dynamic system in the parameter B 0 .
The dynamics of the simplest torus quantum vortex knots in a superfluid at zero temperature has been simulated with a regularized Biot–Savart law (the torus radii R 0 and r 0 for the initial vortex configuration are much larger than the width of the vortex core ). The evolution times of knots until their significant deformation have been calculated with a small step in the parameter B 0 = r 0 / R 0 for different values of the parameter Λ = log( R 0 /ξ). It has been found that regions of quasi-stability appear at Λ ≳ 3 in the range B 0 ≲ 0.2, which correspond to long knot lifetimes and very large traveling distances up to several hundred R 0 . This result is new and quite surprising because previously it was believed that the maximum lifetime of torus knots until reconnection does not exceed several typical periods. The opening of quasi-stable “windows” at increasing Λ is due to narrowing of main parametric resonances of the dynamic system in the parameter B 0 .
The dynamics of torus vortex configurations V n, p, q in a superfluid liquid at zero temperature ( n is the number of quantum vortices, p is the number of turns of each filament around the symmetry axis of the torus, and q is the number of turns of the filament around its central circle; radii R 0 and r 0 of the torus at the initial instant are much larger than vortex core width ξ) has been simulated numerically based on the regularized Biot–Savart law. The lifetime of vortex systems till the instant of their substantial deformation has been calculated with a small step in parameter B 0 = r 0 /R 0 for various values of parameter Λ = ln( R 0 /ξ). It turns out that for certain values of n, p , and q , there exist quasi-stability regions in the plane of parameters ( B 0 , Λ), in which the vortices remain almost invariable during dozens and even hundreds of characteristic lengths.
The dynamics of the simplest vortex knots, “unknots,” and torus links in an atomic Bose condensate at zero temperature in an anisotropic harmonic trap has been simulated numerically within the three-dimensional Gross–Pitaevskii equation. It has been found that such quasistationary rotating vortex structures exist for a very long time in wide ranges of the parameters of the system. This new result is qualitatively consistent with a previous prediction based on a simplified one-dimensional model approximately describing the motion of knotted vortex filaments.
The long-time existence of topologically nontrivial configurations of quantum vortices in the form of torus knots and links in trapped Bose-Einstein condensates is demonstrated numerically within the three-dimensional Gross-Pitaevskii equation with an external anisotropic parabolic potential. We identify parametric domains of trap anisotropy, characterized by the axial over planar frequency ratio $\ensuremath{\lambda}\ensuremath{\approx}1.5--1.6$, where the lifetime of such quasistationary rotating vortex structures is many hundreds of typical rotation times. This suggests the potential experimental observability of the structures. We quantify the relevant lifetimes as a function of the model parameters (e.g., $\ensuremath{\lambda}$) and initial condition parameters of the knot profile.
The dynamics of a quantum vortex toric knot TP,Q and other analogous knots in an atomic Bose condensate at zero temperature in the Thomas–Fermi regime is considered in the hydrodynamic approximation. The condensate has a spatially inhomogeneous equilibrium density profile ρ(z, r) due to the action of an external axisymmetric potential. It is assumed that z*= 0, r*= 1 is the point of maximum of function rρ(z, r), so that δ(rρ) ≈ –(α–)z2/2–(α + )(δr)2/2 for small z and δr. The geometrical configuration of a knot in the cylindrical coordinates is determined by a complex 2πP-periodic function A(ϕ, t) = Z(ϕ, t) + i[R(ϕ, t))–1]. When |A| ≪ 1, the system can be described by relatively simple approximate equations for P rescaled functions Wn(φ)∝A(2πn+φ):iWn,t=−(Wn,φφ+αWn−∈Wn∗)/2−∑j=n1/(Wn∗−Wj∗) . For = 0, examples of stable solutions of type W n = θ n (ϕ–γt)exp(–iωt) with a nontrivial topology are found numerically for P = 3. In addition, the dynamics of various unsteady knots with P = 3 is modeled, and the tendency to the formation of a singularity over a finite time interval is observed in some cases. For P = 2 and small ≠ 0, configurations of type W0–W1 ≈ B0exp(iζ) + C(B0, α)exp(–iζ) + D(B0, α)exp(3iζ), where B0 > 0 is an arbitrary constant, ζ = k0ϕ–Ω0t + ζ0, k0 = Q/2, and Ω0 = (–α)/2–2/B02, which rotate about the z axis, are investigated. Wide stability regions for such solutions are detected in the space of parameters (α, B0). In unstable zones, a vortex knot may return to a weakly excited state.
Lord Kelvin proposed that atoms form hydrodynamic vortex knots. However, they typically untie through reconnections, i. e., local cut-and-slice events, unlike stable vortex unknots such as smoke rings. The same holds in superfluids--quantum fluids with zero viscosity--where vortices have quantized circulation, making them topologically stable. For over 150 years, hydrodynamically stable vortex knots have been sought both experimentally and theoretically. Here, we present the first demonstration of hydrodynamically stable vortex knots and links in experimentally realizable Bose-Einstein condensates of ultracold atomic gases and confirm it through dynamic simulations. Our method creates stable knotted vortex structures in systems where reconnections are prohibited, with potential relevance to neutron star interiors. Additionally, we anticipate our mathematical framework could have applications in quantum computation, quantum turbulence, and DNA dynamics, particularly where reconnections are restricted.
We experimentally study the dynamics of quantum knots in a uniform magnetic field in spin-1 Bose-Einstein condensates. The knot is created in the polar magnetic phase, which rapidly undergoes a transition toward the ferromagnetic phase in the presence of the knot. The magnetic order becomes scrambled as the system evolves, and the knot disappears. Strikingly, over long evolution times, the knot decays into a polar-core spin vortex, which is a member of a class of singular SO(3) vortices. The polar-core spin vortex is stable with an observed lifetime comparable to that of the condensate itself. The structure is similar to that predicted to appear in the evolution of an isolated monopole defect, suggesting a possible universality in the observed topological transition.